Let X1, X2, …, Xn denote a random sample of size n>1 from a...

Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample of size $n>1$ from a distribution with pdf $f(x ; \theta)=\theta e^{-\theta x}, 0<x<\infty$, zero elsewhere, and $\theta>0 .$ Then $Y=\sum_{1}^{n} X_{i}$ is a sufficient statistic for $\theta$. Prove that $(n-1) / Y$ is the MVUE of $\theta$.

Key Concepts

Exponential Distribution

This distribution is characterized by a constant hazard rate and the memoryless property. Its probability density function is commonly expressed as f(x;?)=?e^(??x) for x > 0 and ? > 0, and it is widely used in reliability and survival analysis. Its properties facilitate analytic derivations for sums of independent observations, resulting in gamma distributions.

Sufficient Statistic

A statistic is called sufficient for a parameter if it captures all the information present in the sample about that parameter. For the exponential distribution, the sum of the sample values is a sufficient statistic for ? by the Factorization Theorem, meaning that no additional data from the sample can provide further insight into the value of ? once this sum is known.

Completeness

A statistic is complete if no non-zero function of it has an expectation equal to zero for all values of the parameter. Completeness strengthens the usefulness of sufficient statistics because, in the presence of completeness, unbiased estimators that are functions of the statistic are unique. In this context, the statistic (the sum of the sample values) is complete, ensuring that any unbiased estimator based solely on it is the MVUE.

Minimum Variance Unbiased Estimator (MVUE)

The MVUE is an estimator that is unbiased and has the lowest variance among all unbiased estimators for a parameter. It is sought after in estimation theory because it provides the best linear unbiased estimate in terms of variance. Establishing that an estimator is unbiased based on a complete sufficient statistic guarantees its optimality through uniqueness.

Lehmann-Scheffé Theorem

This theorem states that if a statistic is complete and sufficient for a parameter, any estimator that is an unbiased function of that statistic is the unique MVUE of the parameter. In problems like this, where the sufficient statistic from an exponential family is complete, the application of the Lehmann-Scheffé theorem confirms that the proposed estimator is indeed the MVUE.

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